Sum of Series Calculator
Solve sum series problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Sum of Series Calculator
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Formula: Arithmetic: S = n/2(a+l) | Geometric: S = a(1-r^n)/(1-r)
Worked example — Sum = 670 | Last term = 62 | Average term = 33.5
Formula
Arithmetic: S = n/2(a+l) | Geometric: S = a(1-r^n)/(1-r)
For arithmetic series, S = n/2 times (first + last term). For geometric series, S = a(1 - r^n)/(1 - r) where r is the common ratio. For convergent infinite geometric series (|r| < 1), S = a/(1 - r).
Worked Examples
Example 1: Arithmetic Series Sum
Problem:Find the sum of the first 20 terms of the arithmetic series 5 + 8 + 11 + 14 + ...
Solution:a = 5, d = 3, n = 20 Last term l = a + (n-1)d = 5 + 19(3) = 62 S = n/2 x (a + l) = 20/2 x (5 + 62) = 10 x 67 = 670
Result:Sum = 670 | Last term = 62 | Average term = 33.5
Example 2: Infinite Geometric Series
Problem:Find the sum of the infinite geometric series 12 + 6 + 3 + 1.5 + ...
Solution:a = 12, r = 0.5 (|r| < 1, so the series converges) Finite sum of 10 terms: S10 = 12(1 - 0.5^10)/(1 - 0.5) = 12(0.999023)/0.5 = 23.977 Infinite sum: S = a/(1-r) = 12/(1-0.5) = 12/0.5 = 24
Result:Finite sum (10 terms) = 23.977 | Infinite sum = 24
Frequently Asked Questions
What is a mathematical series and how is it different from a sequence?
A sequence is an ordered list of numbers following a pattern, while a series is the sum of the terms in a sequence. For example, 1, 3, 5, 7, 9 is a sequence; 1 + 3 + 5 + 7 + 9 = 25 is the corresponding series. Sequences describe individual terms, while series describe cumulative totals. A finite series sums a specific number of terms, while an infinite series sums infinitely many terms and may or may not converge to a finite value. The distinction matters because a sequence can have well-defined terms without its series converging. Understanding both concepts is fundamental to calculus, analysis, and applied mathematics.
What is an arithmetic series and what is its sum formula?
An arithmetic series is the sum of terms in an arithmetic sequence, where each term differs from the previous by a constant called the common difference (d). The sum of n terms is S = n/2 times (first term + last term), equivalently S = n/2 times (2a + (n-1)d). This formula works because pairing the first term with the last, the second with the second-to-last, and so on produces equal pairs. The famous story of young Gauss summing 1 to 100 exemplifies this: 50 pairs each summing to 101 gives 5050. Arithmetic series appear in financial calculations like total payments on a uniformly increasing schedule and in physics for uniformly accelerated motion.
What is a geometric series and when does it converge?
A geometric series is the sum of terms in a geometric sequence, where each term is multiplied by a constant ratio (r) to get the next term. The finite sum of n terms is S = a(1 - r^n) / (1 - r), where a is the first term. An infinite geometric series converges (has a finite sum) only when the absolute value of r is less than 1, in which case the sum equals a / (1 - r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 1 / (1 - 0.5) = 2. When the absolute value of r is 1 or greater, the infinite series diverges. Geometric series are fundamental in finance (present value of annuities), physics (bouncing ball total distance), and signal processing.
What is a harmonic series and why is it important?
The harmonic series is the sum 1 + 1/2 + 1/3 + 1/4 + 1/5 + ..., where each term is the reciprocal of a natural number. Despite its terms approaching zero, the harmonic series diverges (grows without bound), though extremely slowly. After summing one million terms, the total is only about 14.39. This counterintuitive result is important because it shows that terms going to zero is necessary but not sufficient for series convergence. The harmonic series appears in probability (coupon collector problem), music theory (overtone frequencies), and computer science (analysis of algorithms). The partial sums of the harmonic series approximate ln(n) + 0.5772 (Euler-Mascheroni constant).
How do you determine if an infinite series converges or diverges?
Several tests help determine convergence. The divergence test says if terms do not approach zero, the series diverges. The ratio test compares consecutive terms: if the limit of |a(n+1)/a(n)| is less than 1, the series converges absolutely. The root test checks if the nth root of |a(n)| has a limit less than 1. The comparison test compares to a known series. The integral test connects series convergence to improper integrals. The alternating series test applies when signs alternate and terms decrease toward zero. For geometric series, convergence requires |r| < 1. No single test works for all series, and choosing the right test is a key skill in calculus courses.
What are power series and how are they used?
A power series is an infinite series of the form sum of a_n times x^n, where x is a variable and a_n are coefficients. Every power series has a radius of convergence R, within which it converges and outside which it diverges. Taylor and Maclaurin series are power series that represent functions. For example, e^x = 1 + x + x^2/2! + x^3/3! + ..., sin(x) = x - x^3/3! + x^5/5! - ..., and 1/(1-x) = 1 + x + x^2 + x^3 + ... for |x| < 1. Power series are essential in physics for approximating solutions to differential equations, in numerical computing for evaluating transcendental functions, and in engineering for analyzing systems near equilibrium.
What is a telescoping series?
A telescoping series is one where most terms cancel when the sum is written in expanded form, leaving only a few terms at the beginning and end. For example, sum of 1/(n(n+1)) from n=1 to N can be written as sum of (1/n - 1/(n+1)), which telescopes to 1 - 1/(N+1) = N/(N+1). When expanded: (1-1/2) + (1/2-1/3) + (1/3-1/4) + ..., each positive term cancels with the negative term of the next pair. Telescoping series are valuable because they provide exact closed-form sums without approximation. Recognizing a telescoping structure often involves partial fraction decomposition, making it a useful technique in calculus and competition mathematics.
What is the difference between absolute and conditional convergence?
A series converges absolutely if the series of absolute values of its terms also converges. It converges conditionally if it converges but the absolute value series diverges. The alternating harmonic series 1 - 1/2 + 1/3 - 1/4 + ... converges conditionally to ln(2), but the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges. This distinction matters because absolutely convergent series can be rearranged in any order without changing the sum, while conditionally convergent series can be rearranged to converge to any desired value (Riemann rearrangement theorem). Absolute convergence is therefore a stronger and more reliable form of convergence for practical applications.
How are series used in financial mathematics?
Series are fundamental to financial calculations. The present value of an annuity (regular payments) is a finite geometric series with ratio 1/(1+r), where r is the interest rate per period. The formula PV = PMT times (1 - (1+r)^(-n)) / r derives directly from summing the geometric series. Perpetuities (infinite annuities) use the infinite geometric series formula: PV = PMT / r. Growing annuities involve geometric series with ratio (1+g)/(1+r). Bond pricing sums discounted coupon payments plus discounted face value. Mortgage amortization tables are built using series formulas. Understanding these series relationships is essential for financial analysts, actuaries, and anyone making investment decisions.
What are some famous series in mathematics and their sums?
Several infinite series have celebrated exact sums. The Basel problem, solved by Euler in 1734, shows 1 + 1/4 + 1/9 + 1/16 + ... = pi^2/6. The Leibniz formula gives 1 - 1/3 + 1/5 - 1/7 + ... = pi/4. The geometric series 1 + 1/2 + 1/4 + ... = 2. The Grandi series 1 - 1 + 1 - 1 + ... has the Cesaro sum 1/2 but does not converge in the usual sense. The Riemann zeta function generalizes power reciprocal sums: zeta(2) = pi^2/6, zeta(4) = pi^4/90. Ramanujan discovered extraordinary series like 1 + 2 + 3 + ... = -1/12 in a regularized sense, which appears in string theory and the Casimir effect.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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